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Author: Tinku Tara

53-log-x-7-x-x-

Question Number 76034 by hmamarques1994@gmail.com last updated on 22/Dec/19 $$\: \\ $$$$\:\mathrm{53}^{\boldsymbol{\mathrm{log}}_{\boldsymbol{\mathrm{x}}} \left(\mathrm{7}\right)} \:=\:\sqrt{\boldsymbol{\mathrm{x}}} \\ $$$$\: \\ $$$$\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$$$\: \\ $$ Answered by MJS…

1-2-2-3-3-4-4-5-17-18-

Question Number 10493 by ABD last updated on 14/Feb/17 $$\frac{\mathrm{1}}{\mathrm{2}!}+\frac{\mathrm{2}}{\mathrm{3}!}+\frac{\mathrm{3}}{\mathrm{4}!}+\frac{\mathrm{4}}{\mathrm{5}!}+…+\frac{\mathrm{17}}{\mathrm{18}!}=? \\ $$ Answered by mrW1 last updated on 14/Feb/17 $${since}\:\frac{{n}}{\left({n}+\mathrm{1}\right)!}=\frac{\mathrm{1}}{{n}!}−\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)!} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}!}+\frac{\mathrm{2}}{\mathrm{3}!}+\frac{\mathrm{3}}{\mathrm{4}!}+\frac{\mathrm{4}}{\mathrm{5}!}+…+\frac{\mathrm{17}}{\mathrm{18}!} \\ $$$$=\left(\frac{\mathrm{1}}{\mathrm{1}!}−\frac{\mathrm{1}}{\mathrm{2}!}\right)+\left(\frac{\mathrm{1}}{\mathrm{2}!}−\frac{\mathrm{1}}{\mathrm{3}!}\right)+\left(\frac{\mathrm{1}}{\mathrm{3}!}−\frac{\mathrm{1}}{\mathrm{4}!}\right)+\centerdot\centerdot\centerdot+\left(\frac{\mathrm{1}}{\mathrm{17}!}−\frac{\mathrm{1}}{\mathrm{18}!}\right) \\…

The-fifth-nineth-sixteenth-terms-of-a-linear-sequence-are-consecutive-terms-of-an-exponential-sequence-1-Find-the-common-difference-of-the-linear-sequence-in-terms-of-the-first-term-2-Show-t

Question Number 10489 by Saham last updated on 13/Feb/17 $$\mathrm{The}\:\mathrm{fifth},\:\mathrm{nineth},\:\mathrm{sixteenth}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{a}\:\mathrm{linear}\: \\ $$$$\mathrm{sequence}\:\mathrm{are}\:\mathrm{consecutive}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{an}\:\mathrm{exponential} \\ $$$$\mathrm{sequence}\:. \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{common}\:\mathrm{difference}\:\mathrm{of}\:\mathrm{the}\:\mathrm{linear}\: \\ $$$$\mathrm{sequence}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{term} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Show}\:\mathrm{that}\:\mathrm{21}^{\mathrm{th}} ,\:\mathrm{37}^{\mathrm{th}} ,\:\mathrm{65}^{\mathrm{th}} \:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{linear} \\ $$$$\mathrm{sequence}\:\mathrm{are}\:\mathrm{consecutive}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{an}\:\mathrm{exponential}…

An-exponential-sequence-of-positive-terms-and-a-linear-sequence-have-the-same-first-term-the-sum-o-their-first-term-is-3-the-sum-of-their-second-term-is-3-2-and-the-sum-of-their-third-term-is-6-

Question Number 10488 by Saham last updated on 13/Feb/17 $$\mathrm{An}\:\mathrm{exponential}\:\mathrm{sequence}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{terms}\:\mathrm{and}\:\mathrm{a} \\ $$$$\mathrm{linear}\:\mathrm{sequence}\:\mathrm{have}\:\mathrm{the}\:\mathrm{same}\:\mathrm{first}\:\mathrm{term}.\:\mathrm{the}\:\mathrm{sum} \\ $$$$\mathrm{o}\:\mathrm{their}\:\mathrm{first}\:\mathrm{term}\:\mathrm{is}\:\mathrm{3},\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{their}\:\mathrm{second}\:\mathrm{term} \\ $$$$\mathrm{is}\:\frac{\mathrm{3}}{\mathrm{2}},\:\mathrm{and}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{their}\:\mathrm{third}\:\mathrm{term}\:\mathrm{is}\:\mathrm{6}.\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{sum}\:\mathrm{of}\:\mathrm{their}\:\mathrm{fifth}\:\mathrm{term}. \\ $$ Answered by mrW1 last updated…